MA005 Study Guide

Unit 4: Derivatives and Graphs

4a. State whether a given point on a graph is a global/local maximum/minimum

  • What distinguishes a global maximum from a local or relative maximum?
  • What distinguishes a global minimum from a local or relative minimum?

Global extreme values (extrema) are either larger (maximum) or smaller (minimum) than any value on a function. Local or relative extreme values are larger (maximum) or smaller (minimum) than and nearby value on a function. See Figure 4a.1. If the derivative of a function at a point is equal to some finite positive or negative value, then it is not an extreme value, either global or local.

Figure 4a.1.

Figure 4a.1.

To review, see Finding Maximums and Minimums.

 

4b. Find critical points and extreme values (max/min) of functions by using derivatives

  • What conditions must be met for a number to be a critical number (critical point)?
  • What does the Extreme Value Theorem say about finding global extreme values on a closed interval?

Critical numbers are values in the domain of a function where the derivative at that point is either equal to zero or undefined. These values are candidates for identifying possible locations of extrema. If the interval is open, then these are the only possible locations for extreme values, and it is possible that there are no extreme values.

If the interval is closed, then the endpoints are included in the list of critical numbers, and according to the Extreme Value Theorem, the function must attain its global maximum and minimum at some point in the closed interval.

To review, see Finding Maximums and Minimums.

 

4c. Determine the values of a function guaranteed to exist by Rolle's Theorem and by the Mean Value Theorem

  • What is Rolle's Theorem? What conditions must be satisfied?
  • What is the Mean Value Theorem? What conditions must be satisfied and what does it guarantee?

Rolle's Theorem says that if a function has the same value at both endpoints of an interval, and if the function is continuous, then there must be at least one point in the interval where the derivative is zero. In order to apply Rolle's Theorem, you must check both the values at the endpoints (to see that they agree) and test that the function is continuous on the given interval. See Figure 4c.1.

Figure 4c.1.
Figure 4c.1.

The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then there is at least one point on the open interval where the slope of the tangent line to the curve is the same as the slope of the secant line connecting the endpoints of the interval. See Figure 4c.2.

Figure 4c.2.

Figure 4c.2.

To review, see The Mean Value Theorem and Its Consequences.

 

4d. Use the graph of f(x) to sketch the shape of the graph of f'(x)

  • When the graph of a function is increasing, what do we know about the derivative of the function?
  • When the graph of a function is decreasing, what do we know about the derivative of the function?
  • When the graph of a function is constant, what is the derivative of the function?
  • If the graph of a function has a cusp, what do we know about the derivative of the function?
  • If the graph of a function has an extreme value, what do we know about the derivative of the function?
  • Define monotonic.

We can say important things about the derivative of a function and use that information to sketch the derivative function even if we do not have an equation. If the original function is increasing, then we know that the derivative is positive.

Likewise, if the original function is decreasing, then we know the derivative is negative. If the function is constant or has an extreme value, then the derivative is zero. If the derivative has a cusp, then there is a break in the derivative. See Figures 4d.1. and 4d.2.

Figure 4d.1.

Figure 4d.1.

Figure 4d.2.

Figure 4d.2.

A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A function is monotonic if its first derivative (which need not be continuous) does not change sign.

To review, see The First Derivative and the Shape of a Function f(x).

 

4e. Use the values of f′(x) to sketch the graph of f(x) and state whether f(x) is increasing or decreasing at a point

  • When the derivative of a function is zero, what do we know about the original function?
  • When the derivative of a function is undefined, what do we know about the original function?
  • When the derivative of a function is positive, what do we know about the original function?
  • When the derivative of a function is negative, what do we know about the original function?
  • If the derivative of a function is zero at a point, how do we determine if the point is an extreme value or something else?

When the derivative of a function is zero, it corresponds to a critical point, which could indicate an extreme value. If it is zero over an interval, then the graph is constant over that interval.

If the derivative of the function is positive, then the function is increasing. If the derivative of the function is negative, then the original function is decreasing. At a minimum, the derivative is decreasing on the left and increasing on the right of the point. At a maximum, the derivative is increasing on the left and decreasing on the right. See Figure 4e.1.

Figure 4e.1.

Figure 4e.1.

To review, see The First Derivative and the Shape of a Function f(x).

 

4f. Use the values of f'(x) to determine the concavity of the graph of f(x)

  • What is concavity?
  • Draw a section of a graph that is concave down. Concave up?
  • What happens to the value of the second derivative when the graph is concave up? Down?
  • What is an inflection point?
  • What is the value of the second derivative at an inflection point?

Concavity is a way of describing the way in which a graph curves. The graph curves facing upward like a bowl when the second derivative is positive. The graph curves downward like a hill when the second derivative is negative. See Figure 4f.1. When the direction of the curve changes from up to down, this is called an inflection point, and at that point, the second derivative is zero.

Figure 4f.1.

Figure 4f.1.

To review, see The First Derivative and the Shape of a Function f(x).

 

4g. Use the graph of f(x) to determine if f'(x) is positive, negative, or zero

  • What is the sign of the second derivative on the interval (−∞, 0) on the graph of f(x)=1/x?
  • What is the sign of the second derivative on the interval (−∞, ∞) on the graph of f(x)=x2?
  • What is the value of the second derivative at the point x=0 on the graph of f(x)=x3? What is the name of this point?

When the graph curves upward like a bowl, the second derivative is positive. When the graph curves downward like a hill, the second derivative is negative. When the direction changes, the point at which the change happens is called an inflection point, and the second derivative there is zero. See Figure 4g.1.

Figure 4g.1.
Figure 4g.1.

To review, see The First Derivative and the Shape of a Function f(x).

 

4h. Solve maximum and minimum problems by using derivatives

  • What is the first derivative test?
  • What is the second derivative test?
  • If a point represents a local maximum, what do we know about the first and second derivatives at that point?
  • If a point represents a local minimum, what do we know about the first and second derivatives at that point?

In order to solve a problem that involves extreme values like maxima or minima, first set up the equation that models the problem. Find the first derivative and set it equal to zero to determine the critical values that could be candidate solutions. Determine if any solutions can be eliminated based on problem restrictions.

Then use the first derivative test, or the second derivative test to determine if the point represents a maximum or minimum. If the second derivative is zero, the test is inconclusive. Diagrams are often useful for setting up the initial equation to be solved.

To review, see Applied Maximum and Minimum Problems.

 

4i. Restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer

  • What are some possible units of volume? How do you determine which to use from the problem?
  • What are some common geometry formulas that can guide you in setting up equations?
  • What are some useful techniques for organizing information in a problem?
  • What are some techniques for identifying the correct solution if the solution process produces multiple possible solutions?

When solving an applied problem, break down the verbal description of the problem one phrase at a time. It can be helpful to draw a diagram, particularly for geometry problems. It may also help to make a table of values and identify any formulas you know that seem to apply to the problem.

Test multiple solutions to determine which is the maximum or minimum and report that value that was requested by the problem. Check units. Do unit conversions before writing down your equations, so that all the values measuring similar things have the same units (do not mix inches with feet).

To review, see Applied Maximum and Minimum Problems.

 

4j. Determine asymptotes of a function by using limits

  • What kind of asymptote is associated with a limit as x goes to infinity if the limit is not itself infinity?
  • What kind of asymptote is associated with a limit as x goes to a finite value when the limit goes to infinity?
  • What are some of the algebraic tricks that can be used to find the value of a limit at infinity?
  • What are some examples of functions that don't have limits at infinity?

If the limit of a function goes to a finite value as x approaches infinity (or negative infinity), this limiting value represents a horizontal asymptote.

A function can have more than one horizontal asymptote. A function that has an infinite limit (or a one-sided infinite limit) at a point has a vertical asymptote. Asymptotes that are non-constant can be found through long division of rational functions, which may result in linear or non-linear asymptotes.

Linear asymptotes that are not vertical or horizontal may be referred to as slant or oblique asymptotes. Some functions, such as oscillating ones, do not converge to a single value as x increases (or decreases), and therefore have no limit at infinity.

One useful way of finding limits at infinity for a rational function is to divide every term in both the numerator and the denominator by the largest power term in the entire expression, then determine the limit of each term at infinity using properties of limits.

To review, see Infinite Limits and Asymptotes.

 

4k. Determine the values of indeterminate form limits by using derivatives and L'Hôpital's Rule

  • Give some examples of indeterminant forms for limits?
  • What kinds of indeterminant forms does L'Hôpital's Rule apply to?
  • What are some tricks to convert other indeterminant forms into one that L'Hôpital's can be applied to?

L'Hôpital's Rule states that you can find the limit of the form \frac{\infty}{\infty} or \frac{0}{0} by taking the derivative of the numerator and the denominator functions separately, and then re-evaluating the limit for \frac{f'(x)}{g'(x)} at the same point.

The process can be repeated as often as needed until a determinant form is obtained. When another kind of indeterminant form is obtained, they must be rewritten into one of the above forms, such as turning a product into a ratio by moving the reciprocal of a product into the denominator, or when an exponent is involved, by taking the logarithm of the entire expression.

To review, see L'Hopital's Rule.

 

Unit 4 Vocabulary

  • Asymptote
  • Closed interval
  • Concave down
  • Concave up
  • Concavity
  • Critical number (or critical point)
  • Decreasing
  • Extreme Value Theorem
  • Extreme value and an extremum (or multiple extrema)
  • First derivative test
  • Global (absolute) maximum
  • Global (absolute) minimum
  • Horizontal asymptote
  • Increasing
  • Indeterminate form
  • Infinite limit
  • Inflection point
  • L'Hôpital's rule
  • Limit
  • Linear asymptote
  • Local maximum
  • Local minimum
  • Mean value theorem
  • Monotonic
  • Nonlinear asymptote
  • Oblique asymptote
  • Relative maximum
  • Relative minimum
  • Rolle's Theorem
  • Second derivative test
  • Slant asymptote
  • Vertical asymptote